The Analytical Theory of Heat
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The comparison then of these series furnishes the follow ing results. If we divide the circumference 2?r into n equal parts, and take an arc u composed of an integral number p of these parts, and mark the ends of the arcs u, 2u, 3u, ... (n l)u, it follows from the known properties of trigonometrical quantities that the quantities sin Qu, sin lu, sin 2u, sin 3w, ... Sin (n l)u, 252 THEORY OF HEAT. [CHAP. IV.
or indeed cos Ou, cos Iw, cos 2w, cos 3u, ... Cos (n 1) u, form a recurring periodic ser...ies composed of n terms : if we com- 27T pare one of the two series corresponding to an arc u or p.
n with a series corresponding to another arc v or v, and multiply term by term the two compared series, the sum of the products will be nothing when the arcs u and v are different. If the arcs u and v are equal, the sum of the products is equal to |-/?, when we combine two series of sines, or when we combine two series of cosines ; but the sum is nothing if we combine a series of sines with a series of cosines.
What to read after The Analytical Theory of Heat?
You can find similar books in the "Read Also" column, or choose other free books by Jean Baptiste Joseph Fourier to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The comparison then of these series furnishes the follow ing results. If we divide the circumference 2?r into n equal parts, and take an arc u composed of an integral number p of these parts, and mark the ends of the arcs u, 2u, 3u, ... (n l)u, it follows from the known properties of trigonometrical quantities that the quantities sin Qu, sin lu, sin 2u, sin 3w, ... Sin (n l)u, 252 THEORY OF HEAT. [CHAP. IV.
or indeed cos Ou, cos Iw, cos 2w, cos 3u, ... Cos (n 1) u, form a recurring periodic ser...ies composed of n terms : if we com- 27T pare one of the two series corresponding to an arc u or p.
n with a series corresponding to another arc v or v, and multiply term by term the two compared series, the sum of the products will be nothing when the arcs u and v are different. If the arcs u and v are equal, the sum of the products is equal to |-/?, when we combine two series of sines, or when we combine two series of cosines ; but the sum is nothing if we combine a series of sines with a series of cosines.
What to read after The Analytical Theory of Heat?
You can find similar books in the "Read Also" column, or choose other free books by Jean Baptiste Joseph Fourier to read onlineMoreLess
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